onsider
some increasable number that starts with the cipher 1 followed by zeroes.
To get to the first number that starts with the cipher 2, you have to bring
about an increase of 100% in the things being counted -- or by whatever
phenomenon your number measures. Some amount of time or some quantity
of other resources will be required for that, depending on the rate at
which the number is enabled to grow.

Now, for continuing on to the first cipher 3, you have
to assure a change of only 50%. If exponential growth is going on
at a steady rate, more or less, then less time will go by. Likewise
toward succeeding ciphers, each being reached with equal increments in
absolute amounts but declining over time relative to the size of the growing
number itself.

When your number eventually reaches the cipher 9 followed
by zeroes, only about 10% more growth will take you over the top, giving
you another number that starts with cipher 1, and once again you face a
100% climb to reach cipher 2.

ou
will find a simple Monte Carlo model of Benford's Law right here,
and a technical treatment of the subject at the Mathworld
site, where a derivation is given for the probability that a given cipher
will appear in the first position of a number.

Dr. Mark J. Nigrini, an accounting consultant,...gained
recognition a few years ago by applying a system he devised based on Benford's
Law to some fraud cases in Brooklyn. The idea underlying his system is
that if the numbers in a set of data like a tax return more or less match
the frequencies and ratios predicted by Benford's Law, the data are probably
honest. But if a graph of such numbers is markedly different from the one
predicted by Benford's Law, he said, "I think I'd call someone in for a
detailed audit."

Some people will find that practical -- but for nepherious
reasons.

Epilog

In another context (Prime Numbers
are Odd), correspondents noted that people who spend time studying
numbers are themselves "odd" and received a cryptic response...

In the Land of Integers, among sums you can give
even odds to us odds, but among products, we odds are outnumberd by evens
three-to-one.

...which required some explanation.

An integer operand can be either even or odd, and it takes
two operands to make a sum or a product, in four combinations -- even|even,
even|odd, odd|even, odd|odd. The respective sums will be even,
odd, odd, even; the respective products will be even, even, even,
odd.

Thus, for example, it is an even bet that a pair
of dice will turn up an odd number on a given toss, but that's only because
craps takes sums not products.

Many of the integers in the world result from measurements
or counting, and their non-uniform biases are explored above in Benford's
Law. The rest of the integers in circulation are the consequences
of arithmetic whereby the evens are apparently crowding out the odds.

Which may explain something. Is it not
Tyranny of the Majority that pejorates odd as unusual/strange/eccentric?

Benford's
Law: The Model

The empiricists among us may enjoy demonstrating Benford's
Law with a simple Monte Carlo model using a spreadsheet. The conventions
used here include...

columns designated by letters A, B,...

rows designated by numbers 1, 2...

arithmetic operators +, -, *, /

SUM(range) returns the sum of the entries in a range

RAND() returns a random number between 0 and 1

INT(x) returns the integer part of x

logical operation IF(condition, do this, else do that)

For the formulas below, you will need to create a table with
eleven columns (A...K) and 100 rows, plus some space above and outside
for labels and control parameters.

A3...A102 ~~ x1...x100, a sequence of 100 growing
numbers, limited to six digits.B3...B102 ~~ the first non-zero digit of xNC3...C102 ~~ cells marked with 1 for each appearance
of cipher 1 in xND3...D102 ~~ cells marked with 1 for each appearance
of cipher 2 in xN...

K3...K102 ~~ cells marked with 1 for each appearance of
cipher 9 in xN